Looking at a sequence of zeros and ones, we
often feel that it is not random, that is, it is not plausible as an
outcome of fair coin tossing. Why? The answer is provided by
algorithmic information theory: because the sequence is compressible,
that is, it has small complexity or, equivalently, can be produced by a short
program. This idea, going back to Solomonoff, Kolmogorov, Chaitin,
Levin, and others, is now the starting point of algorithmic
information theory.

The first part of this book is a textbook-style exposition of the
basic notions of complexity and randomness; the second part covers
some recent work done by participants of the “Kolmogorov
seminar” in Moscow (started by Kolmogorov himself in the 1980s)
and their colleagues.

This book contains numerous exercises (embedded in the text) that
will help readers to grasp the material.